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Astron. Astrophys. 359, 998-1010 (2000)

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2. Comparison of methods

Two different approaches to measure the scatter curve are described in the literature. The basic difference between the `single' method described by Bruch (1996) and used successfully in the present study as well as by Bruch et al. (2000), and the `ensemble' method first introduced by Horne & Stiening (1985) and later adopted by Welsh & Wood (1995) and Bennie et al. (1996), is as follows: In the first case the scatter is determined as a function of phase for each light curve of a given system individually. The difference between a smoothed light curve and the original one is taken to define the scatter. The average of many individual curves of this kind then constitutes the mean scatter curve. In the second case the scatter is defined by the deviation of the individual light curves at a given phase from the mean light curve (after trying to account for long term variations). Thus, the final scatter curve resulting from the `single' method is the mean of the scatter, whereas in the `ensemble' method it is the scatter around the mean. These are different things!

Welsh et al. (1996) compare the two methods qualitatively. However, a more rigorous comparison has never been made. Moreover, the `ensemble' method is only described briefly in the references cited above. Therefore, before embarking on the main objective of the present study - the determination of the location of the flickering light source in four systems - a more thorough assessment of the basis of the `ensemble' method and its relationship to the `single' method is performed in this section.

2.1. The `ensemble' method

The `ensemble' method was introduced by Horne & Stiening (1985) who applied it to the nova-like variable RW Tri. In a quite straight forward way they define the rms-light curve as the root-mean-squared deviation between the individual light curves and their mean light curve as a function of phase. Horne & Stiening (1985) found the scatter to drop sharply in a small phase interval around eclipse centre and concluded "that the flickering light source is more compact than the steady disc light" and "that the flickering distribution is approximately centred on the disc".

The simple approach of Horne & Stiening (1985) furnished sensible results because they made sure that all of their light curves were observed within a short interval of just four days and that the object of their study, RW Tri, is a UX UMa type nova-like variable and thus a member of the most stable class of CVs. Both points together minimize the importance of long term variations. Thus, the mean light curve is stable at all orbital phases.

This was different in the case of the study of HT Cas by Welsh & Wood (1995). They used the `ensemble' method to calculate the scatter curve based on light curves which were collected over a time span of several years at different telescopes. Moreover, as a SU UMa type dwarf nova HT Cas is by far not as stable as RW Tri. These systems do not only exhibit flickering but also aften much stronger long term variations and variations on orbital time scales than nova-like variables. Furthermore, there is no guarantee that those variations which are not attributable to flickering influence the entire light curve in the same way. On the contrary, there is ample evidence for variations in such stars which affect only a part of the orbit, e.g. variable hump amplitudes, appearance and disappearance of intermediate humps, or variations of the eclipse depth. Such effects have to be accounted for before the scatter curve of the flickering can be calculated from the differences between the individual and the mean light curves.

Moreover, Welsh & Wood (1995) introduced a bias into their analysis of the HT Cas light curves. They find that the mid-eclipse flux is variable and attribute it to long term secular trends. They remove this effect by applying "a small additive constant to each curve so that the mean flux at mid-eclipse remained constant". This procedure is problematic because by definition it reduces the scatter of the individual light curves around the mean during eclipse minimum (and even minimizes it if the flickering during eclipse is not stronger than at other phases).

In order to assess if this bias is serious or insignificant the method of Welsh & Wood (1995) was applied to the data of HT Cas used in this paper (see Sect. 3), assuming that the long-term behaviour of the system is not altogether different in the data set of Welsh & Wood (1995) and the present one. Since the light curves available here are only expressed in count rates (as opposed to fluxes) they were normalized to the mean out-of-eclipse brightness in order to put them onto a comparable scale. Thus, long-term brightness variations are assumed not to exist, leading to a lower limit of the scatter. Due to strongly variable eclipse depths (with respect to the out-of-eclipse light level) of HT Cas the resulting scatter curve has a maximum during eclipse phases if no correction is applied to the light curves. It turns into a very deep minimum if the data are treated in the same way as Welsh & Wood (1995) did.

Thus, the bias is potentially quite serious. The minimum in the scatter curve of Welsh & Wood (1995) at the eclipse phase is therefore in the first place caused by the method. Even if the flickering really is eclipsed at these phases - as shown in Sect. 4.1 - this is not the primary cause for the minimum found by Welsh & Wood (1995) in their scatter curve. This conclusion can only be avoided if the actual flux at eclipse minimum does not change much, leading to only small additive corrections. But then, the strongly varying eclipse depths seen at least in the present data suggests that Welsh & Wood (1995) would have measured the long-term variations instead of the flickering. Therefore, without knowledge of further details of their work the results of Welsh & Wood (1995) must be regarded with suspicion.

This shows how important - and dangerous - long term variations in the light curves are when applying the `ensemble' method. Bennie et al. (1996) devised a way to overcome this problem and applied it to RW Tri. Unfortunately they published only a very concise and qualitative description of their method. Here, I will try to reconstruct it.

Due to the briefness of the presentation of Bennie et al. (1996) details of the reconstruction may differ from the original method. But this should not affect the basic idea. However, I will introduce one major difference: Bennie et al. (1996) (just as Horne & Stiening 1985 and Welsh & Wood 1995) expressed their light curves in fluxes, implying that they had calibrated them. The archival light curves for the present study (see Sect. 3) - all of them obtained with photon counting devices - are only available in count rates. They were observed at very different epochs with a variety of instruments. Thus, an otherwise desirable transformation into fluxes is unfortunately not possible. At least for objects (and instruments) furnishing a weak signal, Poisson noise is not negligible in the present context. Whereas it is not obvious how to handle this effect easily once the light curves have been transformed into fluxes, the above mentioned disadvantage is - however only slightly - made up for by the easy application of a Poisson noise correction if the light curves are given in count rates. For these reasons the formulation presented here is expressed in terms of count rates and includes a correction for Poisson noise.

In the spirit of the `ensemble' method the scatter as a function of orbital phase [FORMULA] can be expressed as:

[EQUATION]

Let us regard the individual terms of this equation. The index i stands for an individual light curve. [FORMULA] is a "reference count rate". It can be taken as the mean count rate in parts of the light curve i which are not disturbed by features such as an eclipse or orbital hump. In order to account for long term variations Bennie et al. (1996) assumed that all variations not due to flickering scale linearly with the reference count rate , i.e. that a relation exists such that [FORMULA]. Here [FORMULA] is the count rate which one would observe in the absence of flickering. Note that this is the basic assumption of the method! It is important to realize that the slope b of this relation is permitted to be a function of phase. Thus, each part of the light curve may scale in a different (but linear) way with the reference count rate.

The difference between the observed count rate [FORMULA] and the count rate [FORMULA] predicted in the absence of flickering is then interpreted as being due to flickering. The square of this difference [i.e. of the expression enclosed in square brackets in Eq. (1)] is the contribution of light curve i to the variance around the mean light curve at phase [FORMULA]. It is biased by Poisson noise.

The variance [FORMULA] due to the latter effect is equal to the mean count rate at phase [FORMULA]. If the variations in the light curve in a small interval around [FORMULA] are dominated by Poisson noise [FORMULA] can be taken as the mean count rate within this interval. If it is dominated by flickering (i.e. by real variations of the system brightness), [FORMULA] is the best estimate for [FORMULA]. In praxis, this does not make a significant difference, and I assume [FORMULA]. However, this holds only true if the light curve has not been binned in phase. Otherwise, [FORMULA] is decreased by [FORMULA] where n is the number of original data points (if the light curve is expressed in counts per integration time) or time units (if it is expressed in counts per time unit) in a phase bin. Thus, the second term in the curved brackets in Eq. (1) constitutes the correction for Poisson noise.

The total variance at phase [FORMULA] is then the sum of the individual terms divided by [FORMULA], where [FORMULA] is the number of light curves available at phase [FORMULA]. However, it would be misleading to take the straight mean because this would give exaggerated weight to those light curves which were observed at high count rates (e.g. with a larger telescope) because then the absolute numbers of [FORMULA] and [FORMULA] would be large. An equal weight can be assigned to each light curve by dividing its contribution to the total variance by the square of the reference count rate.

The square root of the total variance is finally the scatter at the phase [FORMULA] which is now independent of the actual order of magnitude of the count rates. Somewhat loosely spoken it can be regarded as the mean amplitude of the flickering in terms of the reference count rate. Its absolute value therefore depends on the strength of the flickering relative to the other light sources in the system.

2.2. The `single' method

The `single' method is described in detail by Bruch (1996) who applied it successfully to the dwarf nova Z Cha. The scatter as a function of orbital phase [FORMULA] is defined in this case as:

[EQUATION]

In contrast to the `ensemble' method no mean light curve is calculated here. Instead, the scatter of the data points around a smoothed version (in the sense that variations considered to be due to flickering are removed but others such as eclipses and variations on longer time scales are preserved) of each individual light curve is taken to define the scatter curve.

Let [FORMULA] be the original count rate of light curve i at phase [FORMULA] and [FORMULA] the count rate of the smoothed light curve. Let there be n data points in a small interval [FORMULA] around phase [FORMULA]. The first term under the square root in Eq. (2) is then the variance of the data points of light curve i around the smoothed light curve in the interval [FORMULA].

It is biased by Poisson noise, the contribution of which to the total variance is just the mean of the count rates within [FORMULA] (in contrast to the `ensemble' method I assume here that the light curves are not binned; a corresponding correction would be trivial). Thus, the second term under the square root in Eq. (2) constitutes a correction for Poisson noise, and the square root itself as a function of [FORMULA] is the scatter curve of light curve i.

Before taking the mean of the individual scatter curves it is necessary to normalize them in order not to lend higher weight to light curves which happen to have been observed at higher count rates for instrumental reasons. As normalization factor the average [FORMULA] of the scatter of light curve i over all phases is taken. Finally, the normalized [thus the index n in Eq. (2)] mean scatter curve is the sum over all individual normalized curves, divided by the number [FORMULA] of light curves contributing at phase [FORMULA].

The critical point of the method is the construction of the smoothed light curve. Is is important that the smoothing is homogeneous, i.e. to ensure that no part of the light curve (in particular the eclipse) is smoothed better or worse than the other parts because this would bias the scatter as a function of phase. A homogeneous smoothing can be achieved by binning the original light curve in suitable (fixed) phase bins and then performing a spline interpolation (better suited than a spline fit because it has fewer degrees of freedom and is thus less subject to arbitrariness) between the points of the binned light curve.

The method has two free parameters: (1) The interval [FORMULA] used to calculate the variance between the original and the smoothed light curve defines the resolution of the scatter curve. Of course, a high phase resolution is always desirable but in praxis it is limited by the phase resolution of the original light curves and the acceptable statistical noise in the final scatter curve. (2) The phase interval over which the original light curves are binned before the smoothed light curve is calculated. Obviously the smoothed curve will follow variations on scales longer than the bin width which consequently will not contribute to the scatter. The choice of the appropriate bin width requires a balance between the maximum time scale of the flickering which is to remain detectable and the necessity for the spline to follow well the eclipse profile. The latter requirement sets an upper limit for the permissible bin width. The lack of sensitivity for flickering on time scales longer than the bin width is the main disadvantage of the `single' method 1. Its main advantage, on the other hand, is its robustness against long term variations of the regarded system. Since the scatter of each individual light curve is calculated (before taking the mean) all long term variations are eliminated when the difference curve [FORMULA] is constructed. This is in contrast to the `ensemble' method where variations on long time scales (or even non-repetitive variations on orbital time scales) represent a major difficulty.

2.3. Application to artificial light curves

In order to test the validity of the concepts described in Sects. 2.1 and 2.2 they will first be applied here to artificial light curves before real data are subjected to them. 100 light curves with artificial flickering based on a shot noise model were generated. Their mean count rates varied randomly between 1000 and 2000 (i.e. in a realistic range but not too large for Poisson noise to become negligible). Each light curve contains an eclipse in the phase interval [FORMULA] with a constant residual count rate of 10% of the mean out-of-eclipse count rate. During eclipse ingress ([FORMULA]) the mean count rate drops linearly and the weight of the flickering drops from 1 at [FORMULA] to 0 at [FORMULA] during this interval. Eclipse egress is simulated in an analogous manner at [FORMULA]. An orbital hump was introduced as the positive half of a sine curve in the interval [FORMULA]. For each of the hundred light curves its amplitude is a random number between 0 and the mean count rate in the interval [FORMULA]. Finally, Poisson noise was simulated by adding to each data point a random number drawn from a Gaussian distribution (good enough an approximation for a Poissonian distribution at the assumed count rates) with a mean of 0 and a standard deviation equal to the value of the data point itself. An example of such an artificial light curve is shown in Fig. 1a.

[FIGURE] Fig. 1. a  An example of an artificial flickering light curve with eclipse. b  Scatter curve of an ensemble of 100 artificial light curves calculated with the `ensemble' method. The reference flux was defined as the mean count rate in the range [FORMULA]. c  Same as above with the reference flux defined in the range [FORMULA]. d  Scatter curve of the same ensemble of artificial light curves calculated with the `single' method.

Before applying the `ensemble' method to these data they were binned in intervals of [FORMULA]. The reference flux was taken to be the mean count rate in the interval [FORMULA] (i.e. disregarding eclipse and hot spot). The resulting scatter curve is shown in Fig. 1b. As expected, during eclipse the scatter is reduced to 0. During egress it rises constantly; the giggles are due to the progressively visible flickering light source. After egress, the scatter remains on a more or less constant level. However, what happened at phases before the eclipse? There is a hump reflecting the orbital hump although the simulated flickering strength at these phases is the same as after the eclipse. The answer is simple: During the construction of the light curves the hump amplitude (relative to the mean count rate at phase [FORMULA]) was varied in a manner completely independent of the mean count rate. Thus, the basic assumption of the `ensemble' method is violated, namely that all variations not due to flickering scale linearly with the reference flux. If, on the other hand, the variations of the hump amplitude are considered as flickering, the hump in the scatter curve is perfectly as expected: It then reflects the enhanced scatter due to this additional flickering component.

In order to study the effect of an unsuitable choice of the reference flux the calculations were repeated using as reference flux the mean count rate in the interval [FORMULA], i.e. right on top or the (variable) hump. The results, shown in Fig. 1c, could have been foreseen: The scatter assumes a minimum at the phases where [FORMULA] was defined while it is augmented at other (out-of-eclipse) phases. The eclipse bottom is somewhat elevated, but not enough to significantly alter the properties of the scatter eclipse.

Finally, Fig. 1d shows the scatter curve of the same sample of artificial light curves, calculated with the `single' method. Obviously, the scatter conforms perfectly well with the expectations with the exception of a slightly too small eclipse width. This is an artifact explained by the discontinuous transition between the ingress/egress slopes and the constant eclipse bottom which cannot be followed by the smoothed light curve. The orbital hump is not seen in the scatter curve because it is connected with variations well above the time scale for which the `single' method is sensible. Whether this is desirable or not depends on the point of view: Should orbital variations of the hump amplitude be regarded as flickering or not?

2.4. Application to HT Cas

In the previous section it was shown that in the case or artificial light curves with well defined properties both methods yield the expected results within their respective limitations. Now, the `ensemble' method will be applied to real light curves of the dwarf nova HT Cas (the application of the `single' method to the same data will follow in Sect. 4.1).

Since HT Cas does not show a strong orbital hump, and since the effect of an unsuitable choice of [FORMULA] on the scatter eclipse is limited, the entire light curve with the exception of the eclipses ([FORMULA]) was used to define [FORMULA]. The resulting curve is shown in Fig. 2c. It has no resemblance with the corresponding curve calculated with the `single' method (Sect. 4.1, Fig. 3). In particular, no trace of a scatter eclipse is visible. Quite on the contrary! This can be understood regarding the three exemplary light curves of HT Cas (binned in phase for clarity but not altered concerning their count rates) in Fig. 2b. The count rates out of eclipse are not proportional to those at the minima. Thus, the basic assumption of the `ensemble' method is violated. This is underlined by the relation between the reference count rate and the count rate at eclipse minimum shown in Fig. 2a (here, the data points representing the light curves shown in the middle frame are ringed). It is obvious that both quantities are not proportional to each other but only loosely correlated.

[FIGURE] Fig. 2. a  Count rates of the light curves at HT Cas at eclipse bottom as a function of the out-of-eclipse count rate. The ringed data points correspond to the light curves shown in the next frame. b  Three light curves of HT Cas, binned in phase for clarity but unaltered concerning the count rates. c  Scatter curve of HT Cas calculated using the `ensemble' method.

[FIGURE] Fig. 3. a  Representative light curve of HT Cas. b  Normalized mean orbital light curve of HT Cas. c  Mean scatter curve of HT Cas. The dashed vertical lines indicate the contact phases of the white dwarf eclipse as measured by Horne et al. (1991). A representative error bar is shown in the upper left corner.

In principle the violation of the basic assumption could be due to the fact that the light curves are expressed in count rates rather than in fluxes. As was detailed in Sect. 2.1 the method assumes that long term variations at a given phase scale linearly with the reference count rate. In the present context variations of the count rates may be due to real long term variations as well as to different instrumental setups. The latter would cause the same relative variation at all phases while for the former the relative variations can be phase dependent. Thus, mixing real long term variations and such due to instrumental effects causes a superposition of two different equations of the kind [FORMULA] (see Sect. 2.1), resulting in an apparently enhanced scatter.

Although this effect may contribute to the failure of the `ensemble' method in the present case it is certainly not the principle reason. The strongly variable eclipse depth in HT Cas with respect to the out-of-eclipse light level is independent of the absolute value of the count rates and would clearly lead to a strong scatter during eclipse even if the light curves were expressed in fluxes.

Application of the `ensemble' method to light curves of UX UMa (Sects. 3 and 4.4) led to a similar failure.

2.5. Conclusions

Although the `ensemble' method is quite appealing at first glance, and although its first application by Horne & Stiening (1985) met success, its application is limited to "well behaved CVs". This term, however, is almost a contradiction in itself. As was pointed out by Welsh et al. (1996) the sensitivity of the `ensemble' method to violations of stationarity is a major disadvantage. This does not mean that it cannot yield useful results under carefully chosen and controlled circumstances which, however, for the large majority of CVs will be difficult to realize. In view of the additional difficulties arising from the fact that the light curves available for the present study are all uncalibrated the application of the `ensemble' method to these data is definitely not suitable. Therefore, the `single' method is preferred here.

Doubtlessly, the `single' method also has its drawbacks: It yields rather noisy scatter curves unless the number of light curves is exceedingly high, and it is insensitive to flickering on time scales longer than those of the remaining variations in the smoothed light curves. However, as has been shown by Bruch (1996), Bruch et al. (2000) and will be shown in Sect. 4 of the present paper, it is capable to produce useful scatter curves of the flickering even for the most unstationary CVs.

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© European Southern Observatory (ESO) 2000

Online publication: July 13, 2000
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